## Theory of the Algebraic Functions of a Complex Variable |

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Theory of the Algebraic Functions of a Complex Variable John Charles Fields No preview available - 2015 |

Theory of the Algebraic Functions of a Complex Variable John Charles Fields No preview available - 2015 |

### Common terms and phrases

2n coincidences according to powers adjoint function adjoint numbers adjoint orders adjointness relative algebraic functions arbitrary constants involved basis become infinite birational transformation branches corresponding Chapter cidence cients coeffi coefficients coincidence corresponding complementary adjoint complementary formula complete set corresponding numbers cycle of order cycles corresponding degree dence divisible by z—a element equal evidently expression factor finite values follows fractional func function H function H(z,v function of z,v fundamental algebraic equation fundamental equation given value H. F. Baker independent conditions indicate infinities integral rational function irreducible equations level furnished multiplying negative notation number of arbitrary number of conditions obtain orders of coincidence p-function polynomial Q coincidences q factors reduced form represent requisite to adjointness Riemann-Roch theorem set of adjoint set of coincidences set of numbers set of Q simple branch simultaneously be greater simultaneously greater summation supposed system of functions tion total number vanish identically variable in question zero

### Popular passages

Page 134 - Ni in terms of the new notation represents the number of arbitrary constants involved in the expression of the most general rational function, whose orders of coincidence with the branches of the several cycles corresponding to the various finite values z = ak exceed by — TI*', — .:(2},... — tj...

Page 134 - T{-00' respectively. arbitrary constants involved in the expression of the most general rational function of (z, v), whose orders of coincidence with the branches of the several cycles corresponding to the various finite values z = a,. exceed by ol*', i(2\ . •. o^.

Page 138 - From (2) it is seen that q — s + 1 is the number of arbitrary constants involved in the expression of the most general rational function of (z, «), whose infinities, all of the first order, correspond to points among the q points here in question.

Page 34 - ... are adjoint, we say that the function is adjoint for the value of the variable in question. To say that a rational function of (z, u) is adjoint for a value of the variable z, is evidently equivalent to saying that its orders of coincidence with the branches of the several cycles are greater than...

Page 147 - ... for we have shewn that it is equal to the number of the arbitrary constants involved in the expression of the general integral rational function built on this basis. Employing the notation N? to designate the number of the arbitrary constants involved...

Page 88 - ... as a result of equating to 0 the principal residue in the product (26). On equating to 0 the principal residue in the product (25) we then impose no further conditions on the coefficients ar-.^t~i than those already obtained on equating to 0 the principal residue in the product (21).

Page 126 - What are the necessary and sufficient conditions, which must be satisfied by the coefficients of...

Page 134 - The expression (18) then represents the number of arbitrary constants involved in the expression of the most general rational function...

Page 135 - We may conceive a set of numbers to be associated with each value of the variable, the numbers however being all 0 save in the sets associated with a finite number of values of z. When...

Page 102 - Q — q + ï is the number of arbitrary constants involved in the expression of the most general rational function whose infinities are included under a certain set of Q infinities, Ada matliematira.